2.67 Repeating as a Fraction- Complete Calculation Guide

What Does 2.67 Repeating Actually Mean?

2.67 repeating is the decimal number 2.676767... — the "67" goes on forever. It's not 2.67 rounded. It's not approximately 2.67. It is exactly 2.6767676767... and keeps going.

Mathematicians write this as 2.67 with a bar over the 67, or sometimes with parentheses: 2.67 or 2.(67).

Here's the thing: repeating decimals are rational numbers. They can all be written as fractions. 2.67 repeating is no exception.

The Fraction Conversion: Step by Step

Let's convert 2.676767... to a fraction.

Step 1: Set Up the Equation

Let x equal the repeating decimal:

x = 2.676767...

Step 2: Multiply to Match the Repetition

The repeating part is "67" — that's 2 digits. Multiply both sides by 100 (which is 10²):

100x = 267.676767...

Step 3: Subtract to Eliminate the Decimal

Subtract the original equation from this new one:

100x - x = 267.676767... - 2.676767...

99x = 265

Step 4: Solve for x

x = 265/99

That's the answer. 2.67 repeating as a fraction is 265/99.

Can You Simplify 265/99?

Let's check. Find the prime factors:

• 265 = 5 × 53
• 99 = 9 × 11 = 3² × 11

No common factors. 265/99 is already in lowest terms.

As a mixed number: 2 67/99

How to Verify Your Answer

Divide 265 by 99 to check:

265 ÷ 99 = 2.676767... ✓

The decimal repeats "67" exactly. Your fraction is correct.

Why This Method Works

You're using algebra to cancel out the infinite decimal. Here's the logic:

• You have x = 0.ABABAB... where "AB" is the repeating part
• Multiply by 100 shifts the decimal two places
• Subtracting removes the decimal portion entirely
• You're left with a simple equation to solve

The same process works for any repeating decimal, regardless of how many digits repeat.

The General Formula for Repeating Decimals

For a repeating decimal with a single-digit repeat (like 0.3̄):

x = 0.333...

10x = 3.333...

10x - x = 3

x = 3/9 = 1/3

For two-digit repeats, multiply by 100. For three-digit repeats, multiply by 1000. The multiplier is always 10^(number of repeating digits).

Quick Reference Table

Common Mistakes to Avoid

• Confusing 2.67 with 2.67 repeating. 2.67 = 267/100. 2.6767... = 265/99. Different numbers.
• Using the wrong multiplier. If the repeat is "67", use 100. If it's "7", use 10.
• Forgetting to subtract. The subtraction step is what removes the decimal. Skip it and you're stuck.
• Not simplifying. Always check if your fraction can be reduced further.

How to Get Started: Converting Any Repeating Decimal

Here's the process you can apply to any repeating decimal:

1. Identify the repeating block and count its digits (let's call this n)
2. Multiply the decimal by 10ⁿ
3. Subtract the original number from the result
4. Divide both sides by (10ⁿ - 1)
5. Simplify if possible

Example: 0.1̄6

• Repeating digit is "6" → n = 1
• x = 0.1666...
• 10x = 1.666...
• 10x - x = 1.6
• 9x = 1.6 = 16/10
• x = 16/90 = 8/45

That checks out. 8/45 = 0.1777... wait — let me recalculate. Actually 0.1̄6 = 0.1666... = 1/6 = 15/90 + 1/90... The fraction is 1/6.

The Bottom Line

2.67 repeating as a fraction is 265/99. That's it. No tricks, no rounding, no approximation.

The conversion method works every time: multiply, subtract, divide. The math is reliable. Use it whenever you need to turn a repeating decimal into its exact fractional form.